| import Mathlib |
| import OAI.Probability.Perceptron.Cavity.BulkDiagonalLimit |
| import OAI.Probability.Perceptron.Variational.MarkedGraphLaw |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open MeasureTheory ProbabilityTheory Set Filter |
| open scoped Classical ENNReal NNReal BigOperators Topology BoundedContinuousFunction |
| namespace SphericalPerceptronFreeEnergy |
|
|
| theorem bulk_marked_rpc_coupling (M : ℕ→ℕ) (g : Jet3) (v : ℕ→ℕ→ℝ) |
| (hv : ∀ n p,1≤v n p) {C : ℝ} (hC : 0≤C) (hvC : ∀ n p,|v n (p+1)|≤C) |
| (hdO : ∀ p,Tendsto (fun n=>bulkDeviationAt n (M n) g.f p (v n)) atTop (𝓝 0)) |
| (hdF : ∀ p,Tendsto (fun n=>bulkDeviationAt n (M n+2) g.f p (v n)) atTop (𝓝 0)) |
| (s : ℕ→ℕ) (hs : StrictMono s) (α : ℝ) (hα : 0≤α) |
| (hd : Tendsto (fun n=>((M (s n)+2:ℕ):ℝ)/(s n+1:ℕ)) atTop (𝓝 α)) |
| (ν : ProbabilityMeasure (CompactArray CompactOverlap)) |
| (ho : Tendsto (fun n=>bulkGibbsArrayLaw (s n) (M (s n)) g.f (v (s n))) atTop (𝓝 ν)) |
| (K : ℝ) (hK0 : 0≤K) (hK : ∀ n,((M (s n)+2:ℕ):ℝ)/(s n+1:ℕ)*‖g.d1‖^2≤K) |
| (μ : ProbabilityMeasure (CompactArray (BulkPairRange K))) |
| (hf : Tendsto (fun n=>bulkMarkedArrayLaw (s n) (M (s n)+2) g (v (s n)) K (hK n)) atTop (𝓝 μ)) : |
| ∃ d : ℝ, ∃ hd0 : 0 ≤ d, ∃ hdK : d ≤ K, |
| d ≤ α*‖g.d1‖^2 ∧ |
| let η := markedTimePair K μ |
| let q := boundedQuantile η |
| let hq := boundedQuantile_monotone η |
| let B := (α*‖g.d1‖^2)/(1+α*‖g.d1‖^2) |
| Tendsto (fun j => cavityLabelMarkedLaw |
| (fun i => continuousCavityProfile η B hd0 hdK (roundTimeValue (unitQuantileField q hq) j i)) |
| (cavityTrueDiagonal hd0 hdK) |
| (stepCumulative (strictRoundModel (unitQuantileField q hq) j).weight)) atTop (𝓝 μ) := by |
| obtain ⟨a,ham,hab,hag,hsc,hbound⟩ := |
| bulk_marked_self_consistency M g v hv hC hvC hdO hdF s hs α hα hd ν ho K hK0 hK μ hf |
| obtain ⟨d,hd0,hdK,hdA,hD2,hBd⟩ := |
| bulkMarkedArray_diagonal_producer M g v (Eventually.of_forall hv) hdO s hs α hα hd ν ho K hK μ hf |
| refine ⟨d,hd0,hdK,hdA,?_⟩ |
| let η := markedTimePair K μ |
| let q := boundedQuantile η |
| let hq := boundedQuantile_monotone η |
| let B := (α*‖g.d1‖^2)/(1+α*‖g.d1‖^2) |
| have hA : 0 ≤ α*‖g.d1‖^2 := mul_nonneg hα (sq_nonneg _) |
| have hB : B < 1 := (div_lt_one (by linarith : 0 < 1+α*‖g.d1‖^2)).mpr (by linarith) |
| have he e := compact_exchangeability_limit hf e |
| (fun n => bulkMarkedArray_exchangeable (s n) (M (s n)+2) g (v (s n)) K (hK n) e) |
| have hp := bulkMarkedArray_limit_overlap (fun n => M n+2) g v s K hK hf |
| have hg := bulk_limit_geometry (fun n => M n+2) g.f v (Eventually.of_forall hv) hdF hs hp |
| have hgeom := marked_overlap_geometry K μ he hg.1 hg.2.1 |
| have hD1 := ae_of_ae_map (compactMapArray_continuous (show Continuous |
| (Prod.fst : BulkPairRange K → CompactOverlap) from continuous_fst)).measurable.aemeasurable |
| (bulkGibbsArray_limit_diagonal g.f (fun n => M n+2) v s hp) |
| have hgraph := marked_graph_law μ hB hd0 hdK he hgeom.2.2 hD1 hD2 hBd a |
| (fun r => (hab r).1) hag hsc hbound |
| obtain ⟨hGG,hGe,hEx⟩ := bulk_marked_time_inputs (fun n => M n+2) g v |
| (Eventually.of_forall hv) hdF s hs K hK μ hf |
| have hpair : (markedTimeLaw K μ : Measure (CompactArray Time)).map (fun Q => Q 0 1) = timeLaw.map q := |
| (boundedQuantile_law η).symm |
| have ht := cavityLabelMarkedLaw_tendsto_graph (markedTimeLaw K μ) q hq hpair hGG hGe hEx |
| (continuousCavityProfile η B hd0 hdK) (continuousCavityProfile_continuous η hB hd0 hdK) |
| (cavityTrueDiagonal hd0 hdK) |
| rw [hgraph] at ht |
| exact ht |
|
|
| end SphericalPerceptronFreeEnergy |
|
|
| end |
|
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| end OAI |
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|
